Textures appearing different in Edit and Object mode
You have a Decimate modifier which is removing some geometry in flat areas, messing up the UVs.

Remove (by pressing the X icon) or disable it (by pressing the camera and eye icons) and then everything will work properly:





As for mirroring, delete half of your model and then add a mirror modifier:



Note that the object origin has to be at the point you want to mirror around, so select the central loop and press ShiftS>Cursor to selected, then in object mode press CtrlShiftAltC>Origin to 3D cursor:



What this does is map mirrored faces on top of each other in the UV map (if you apply the modifier you get your complete rabbit and you can see the resulting UVs in the UV editor).